will have only limited accuracy because there is insufficient information left to accurately characterize its shape. Conditions arise in the raw GC/MS data for which it is not possible to extract resolved mass spectra unambiguously. One case is when the elutant-to-background ratio falls significantly below 5%. In these cases, the very weak intensity ions, including isotope ions, usually do not appear in the resolved mass spectra. The other difficulties arise when i t is not possible to detect the presence of multiple elutants because they occur within less than one mass spectrum scan time of each other. In this case, the processed spectrum represents the mixture of the two elutants. In general, we have found that the present system works very well and is capable of detecting and isolating high quality representative mass spectra in GC/MS experiments involving complex biological mixtures.
ACKNOWLEDGMENT We thank D. Smith, W. Pereira, and W. Yeager who have contributed in a major way to the continued refinement of the computer programs implementing these algorithms and to the critique of results from their operational use in our laboratory. We also acknowledge the work of B. E. Blaisdell of Juniata College, Huntingdon, Pa., on an exploratory alternative approach to the present problem. LITERATURE CITED (1) R A Hites and K Biemann, Anal. Chem., 42, 855 (1970) (2) C C Sweeley. N D Young, J F Holland, and S. C Gates, J Chromatogr, 99, 507 (1974)
W. H. McFadden, "Techniques of Combined Gas Chromatography/Mass Spectrometry: Applications in Organic Analysis", Wiley Interscience. London, 1973. J. E. Biller and K. Biemann, Anal. Left., 7 , 515 (1974). R. E. Summons, W. E. Pereira, W. E. Reynolds, T. C. Rindfleisch, and A. M. Duffield. Anal. Chem., 46, 582 (1974). R. N. Stillwell, 22nd Annual ASMS Conference-Mass Spectrometry, Philadelphia, Pa., 1974, p 454. "Mass Spectra of Compounds of Biological Interest", U.S. At. Energy Comm. Rep., No. TID-26553, S . P. Markey, W. G. Urban, and S.P. Levine, Ed. See for example: R. S. Ledley, "Digital Computer and Control Engineering". McGraw-Hill Book Co., New York, 1960, p 742. E. Grushka, M. N. Myets, and J. C. Giddings, Anal. Chem., 42, 21 (1970). C. D. Scott, D. C. Chiicote. and W. W. Pin, Clin. Chem. ( Winston-Salem, N.C.),16, 637 (1970). See for example: S. D. Conte and C. de Boor, "Elementary Numerical Analysis: An Algorithmic Approach", McGraw-Hill Book Co., New York, 1972. p 241 and following. A. Buchs, A. B. Deifino. A. M. Duffield. C. Djerassi, B. G. Buchanan, E. A. Feigenbaum, and J. Lederberg, Helv. Chlm. Acta, 53, 1394 (1970). R. E. Carhart, D. H. Smith, H. Brown, and C. Djerassi, J. Am. Chem. SOC., 97, 5755 (1975). R. G. Dromey, B. G. Buchanan, D. H. Smith, J. Lederberg, and C. Djerassi, J. Org. Chem., 40, 770 (1975).
RECEIVEDfor review September 15,1975. Accepted April 30, 1976. This work was supported by grants (Nos. RR-612 and GM-20832) from the National Institutes of Health and (No. NGR-05-020-632) from the National Aeronautics and Space Administration.
Systematic Errors in Gas Chromatography-Mass Spectrometry Isotope Ratio Measurements D. E. Matthews and J. M. Hayes* Deparfments of Chemistry and Geology, Indiana University, Bloomington, Ind. 4740 1
The mass cycling error, a systematlc bias inherent in isotope ratio measurements carried out by switching ion beams which are time-variable, is systematically explored in a series of trial calculations, and the validity of these calculations is demonstrated experimentally using a typical gas chromatographymass spectrometry system. The effects of varying measurement conditions are considered in terms of chromatographic peak profiles, relative ion-beam Integrationtlmes, delays required for Ion-beam positioning, unidlrectional vs. bidirectional scanning patterns, and chromatographic fractlonatlon of the isotopes. It is concluded that the unidirectionalscanning pattern (ABCABC .) is superior to the bidirectional pattern (ABCCBA .), and that, in the absence of chromatographic fractionation of the isotopes, a bias of less than 1% can be obtained by its use for ten or more cycles on any given gas chromatographic peak. Under these same conditlons, 35 or more observation cycles are requlred to bring the bias below 0.1%.
..
..
Although high precision (10.1% uncertainty) isotope ratio measurements are normally performed using isotope ratio mass spectrometers equipped with dual viscous leak inlets, new and improved techniques of organic analytical mass spectrometry are leading many laboratories to attempt mea-
surements of isotope ratios via gas chromatography-mass spectrometry (GC-MS) techniques. Using GC-MS, extensive sample purification with subsequent combustion to a gas is not required, and submicrogram quantities of compounds can readily be handled. For example, many biomedical assays are now performed using stable isotope labeled internal standards (1).As these and related measurements are developed, it becomes important to study the fundamental limitations on the accuracy and precision of the method. The preeminent limitation is imposed by ion statistics. Inescapably, the number of ions collected constitutes the ultimate limiting factor on the precision of an ion current ratio measurement (2). Quite apart from purely statistical considerations, however, i t can be shown that a second category of fundamental limitations involves systematic errors which are built into the measurement by the pattern of ion beam observation, and which can degrade accuracy. In this report, the effect of observation patterns on GC-MS isotope ratio determinations is analyzed for the common system in which ion beam switching and electron multiplier detection are employed. Figure l a illustrates a case where two masses are being monitored and an erroneous ion current ratio measurement will result. If mass 1 is observed for the first portion of the GC peak, mass 2 is observed for the second portion, and mass 1 is again observed for the remainder, it is visually obvious that ANALYTICAL CHEMISTRY, VOL. 48, NO. 9, AUGUST 1976
1375
ION FLUX
measurement cycle
+I
lntegratlon time
et
measurement order
u
Y
2
3
t,+c t,+ct2++ t
4
t 2+
0 0 0 0 0 0 time
TIME
molecular flux
(b)
y
TIME
Figure 1. Sketches illustrating how the subdivision of a gas chromatographic peak can lead to a systematic error (a)Extreme case (see text). In the terminology of the present model, this situation is characterized by j = 2, 4 = -0.5. ( b )j = 6,4 = 0
the apparent abundance of mass 2 will be artificially enhanced. Although it is not expected that any practical measurement would be made under these conditions, if an obvious error occurs in the case of Figure la, then what about the case of Figure l b , in which the masses are cycled more frequently? Here it is shown that a model system allowing calculation of these errors can be readily developed, and that the ratio differences predicted by these calculations are in good agreement with actual observations.
THEORY Let R* denote the true ratio between two ion beams, 1 and 2, each of which can be represented as a function of time. Using the symbols F1 and F2 to denote the instantaneous ion fluxes, we can write
If both ion beams are derived from the same parent compound, these ion currents will be linearly related to the rate a t which the parent compound enters the ion source. Then, using F ( t ) to denote the rate of sample input, we can write
-
F l ( t ) = k1 F ( t ) and F2(t) = k z F ( t ) a
(4)
or, in terms of areas, aij
= kiA;j
where a,] denotes the integrated ion current of the ith beam in the j t h time interval and A,, represents the integrated sample input (see Figure 2 ) . Following Figure 2 , and using R (no asterisk) to denote the observed ion current ratio, we obtain (a11 + a12 + a d / t l R= (6) (a21 + a22 + azd/tz where t l and t 2 are the incremental observation times for ion beams 1 and 2, respectively. Since the areas under the ion beam curves can be related to the parent population, Equation 6 can be rewritten as 1376
R = kl(Al1 f
ANALYTICAL CHEMISTRY, VOL. 48, NO. 9, AUGUST 1976
A12
+ A13)/tl
(7) kz(A21 + A22 + A 2 3 ) / t z Defining a measurement cycle as the time required to observe once all of the masses of interest plus the time required to prepare to repeat the process, it can be seen that three measurement cycles are completed in Figure 2. In general, for n measurement cycles, Equation 7 becomes
? AiJ/ti
ki
]=I
R=
(2,3)
where k.1 and 122 are constants. I t follows that R* = kJk2. In practice, ion fluxes are integrated over finite time intervals, and, for any arbitary interval, we can write
SFl(t)dt= kljF(t)dt
Flgure 2. Sketches illustrating the definitions adopted in this work The top curve represents the gas chromatographic peak from which ion fluxes FI and F2 are derived. The division of the peak into measurement cycles is shown on the time-line at the top, and the correspondingareas and integrated ion currents are indicated in the relevant peak profiles
k2
5 AzJ/tz
J=
(8)
1
It is of interest to calculate the extent to which R differs from R*. Accordingly, the mass cycling error, AR, is defined as the relative difference between the observed and true ratios R -R* AR=(9) R* Substituting Equation 8 into Equation 9 and recalling R* = k l / k 2 , we obtain: AR
AiJ/ti =j=1
-1
? A2p2
J=1
Note that AR is independent of R*. Using this expression, mass cycling errors can be computed by the appropriate integration of any function chosen to represent a chromatographic peak shape. Because it represents the most fundamental example, a Gaussian peak shape has been used in
n
I
/ 1 \
(a)
TIME
FLUX
ION
I TIME
Figure 3. Examples showing the effect of varying values of q5 (a) j = 3,q5
= 0. (b) j = 3,q5 = -0.5
calculating most of the results presented here. Because ideal peak symmetry is not always obtained in practical situations(!), additional calculations have been based on a nineterm polynomial function which has been fit to a typical tailing GC peak. Details of the integration and summing procedure require some explanation. First, with regard to the number of time intervals obtained within any given GC peak, it can be observed that the use of computer control usually dictates that only complete measurement cycles are obtained. T h a t is, the measurement is not stopped in the middle of the mass observation sequence, and the index j will be equal for all masses. It is, however, possible that the ion flux will fall below the threshold level before the last measurement cycle is completed. For example, in both Figures 3a and 3b, three complete measurement cycles were performed; however, in Figure 3b, the GC peak ended before the completion of the third cycle, providing only 2% measurement cycles of usable data. In fact, the end of the GC peak will only rarely coincide with the end of the last measurement cycle. To take this into account, a measurement cycle fraction, 4, can be defined such that -1 < 0, with the number of cycles of usable data being equal to j +. In Figure 3a, j = 3, = 0, and the peak is symmetrically divided such that AR = 0. But, in Figure 3b, j = 3 , 4 = -% and AR > 0. In order to consider the mass cycling error as a function of j , an average AR (referred to as can be calculated a t each integer measurement cycle value by integrating over I#J
+
+
m)
BCi)=
so -1
.hRO’,+)dr#J
(11)
In the following discussions, the term “mass cycling error” applies to .hRG). Two basic patterns of observation for the desired masses are possible, allowing for two different scanning algorithms. A unidirectional scan could be performed by observing three ion beams, A, B, C in the following order: ABCABC . . . , or a bidirectional scan could be performed using the order: ABCCBA . . . . A different mass cycling error would be expected for each. Although equal integration times are often used to observe the masses monitored, it can be advantageous to use unequal integration times. T o evaluate the effect of unequal integraa factor, K , which relates the relative intetion times on gration times will be defined: t 2 = k t 1 . It is also necessary to consider the effect of time delays on the mass cycling error. Time delays arise from two sources: first, time is required for mass selection and electronic stabi-
=,
Figure 4. An illustration of the effect of chromatographic fraction of
isotopic species The shaded areas indicate the integrated ion currents for species 1 and 2. In this example,P=37%, j = 3 , 4 = 0
lization. Second, when using selected ion monitoring for more than two masses, any ion current ratio computed can be concerned with only one pair of ion beams. The observation of other masses is extraneous to that measurement and can be considered as a time delay. T o calculate the mass cycling error for both the unidirectional and bidirectional scan algorithms, a model using three different delays per measurement cycle (tdl, tdz, t d 3 ) is required. The time delays can be related to the integration time of mass 1by the factors kdl, k d z , k& tdi
= kditl
(12)
tdz
= kdztl
(13)
td3
= kd&l
(14)
Two consecutive measurement cycles for the unidirectional and bidirectional scans would have the measurement orders: Unidirectional: tdl, t l , td2, t 2 , td3, t d l , t l , td2, t 2 , t& Bidirectional: tdl, t i , t d z , t 2 , td3, t d 3 , t 2 , tdz, t i , t d l Of course, one or more of the time delays could be neglible in comparison to the integration times. In Figure 1, for example, k = 1and kdl = k d z = h d 3 = 0. As a practical matter not specifically related to mass spectroscopic details, partial chromatographic resolution of isotopic species should be considered. Two restrictions will be placed on any discussion of chromatographic isotopic fractionation: the GC peak shapes of the two isotopic species will be identical (an assumption made with confidence), and both isotopic species will be observed fully. This second restriction is arbitrary and potentially troublesome. If a GC-MS system determined the starting and stopping points for observation simply by comparing the flux of the more abundant species to some threshold value, it is quite possible that significant fractions of the minor species could be overlooked. Systematic considerations of this kind of error, which is qualitatively different from the mass cycling error, are outside the scope of this investigation. Chromatographic zones due to two isotopic species are shown in Figure 4. The upper profile represents the flux of isotopic species 1,and the lower profile represents the flux of isotopic species 2. The centers of the two zones are separated by a time difference At. The degree of isotopic fractionation, ,R, can be defined as ANALYTICAL CHEMISTRY, VOL. 48, NO. 9, AUGUST 1976
1377
yz COMPUTER
Figure 5. Block diagram of the GC-MS and sample introduction systems
P=;
At
where w is the peak width for t h e chromatographic zone corresponding t o a single isotopic species. W h e n isotopic fractionation occurs (P # 0), Equation 10 cannot be applied directly; instead, a second parent function must be defined such
that
F’(t
+ At) = F(t)
where F’(t) a n d F ( t ) a r e identical in shape but are separated by a time interval At. F ( t ) describes the first species t o come off t h e GC column a n d F’(t) describes t h e second. T h e areas obtained by integration of these functions are equal when t h e limits of integration of A’ are displaced b y At
A’J
= AI
If t h e denominator i n Equation 10 is modified by the substitution of Azj’ for Azj, t h e mass cycling error can be calculated as a function of t h e degree of chromatographic isotopic fractionation.
EXPERIMENTAL
RESULTS AND DISCUSSION
Apparatus. The mass spectrometer system used consists of a Varian 620i minicomputer interfaced to a Varian MAT CH-7 mass spectrometer as described by Schoeller and Hayes (3).As noted in Figure 5, the effluent from a gas chromatographic column is introduced to the mass spectrometer by means of a palladium tube interface ( 4 ) .Samples can also be introduced to the mass spectrometer by means of a nonfractionating viscous flow inlet system (3) or via a standard molecular leak. Samples are injected into the chromatographic column by a system incorporating an 1137 cm3stainless steel exponential dilution flask (EDF) ( 5 )and 2 Valco 8-port valves (Figure 5). Toggling the first 8-port valve supplies the EDF with 1 cm3 of xenon. A 45 ml/min flow of hydrogen through the EDF dilutes the concentration of xenon in the EDF to a level less than 20 nmol/ml. The flow rate is then slowed to -5 ml/min, thus reducing the exponential decay rate, and allowing a number of samples to be taken in the xenon concentration range of interest. The second 8-port valve samples the Xe/H2 mixture in the EDF, obtaining alternately 1cm3 and 0.2 cm3 aliquots. A coiled 3-m x 1-mm i.d. stainless steel tube packed with 110-124 mesh graphitized carbon black ( 6 )functions as the GC column. The column is operated at -23 O C with a 4.0 ml/min flow of hydrogen carrier gas. The GC-MS interface removes virtually all hydrogen carrier gas, keeping the ion source pressure in the low Torr range yet allowing all xenon to pass into the mass spectrometer. The interface is constructed from a 38-cm length of 0.3-mm o.d., 0.076-mm wall, 75/25 palladium/silver alloy tube (NASA, Viking program, flight-qualified) coiled loosely around a 75-watt cartridge heater situated in the middle of an enclosed aluminum cylinder. Air preheated to -400 “C is allowed to flow through the chamber at a rate of -100 ml/min, with the chamber temperature ranging between 240 and 290 OC. Software. After the approximate mass range is set by manual adjustment of the magnetic field strength, the GC-MS isotope ratio program is used to operate the mass spectrometer. When the compound(s) of interest enters the MS via the molecular leak, viscous leak, or GC injection, the program causes the accelerating voltage to be scanned and the centroid locations of all masses in the scanning range to be calculated and stored. From the list of masses located, the operator selects those which are to be monitored and for which abundance ratios are to be computed. After entering a few more parame1378
ANALYTICAL CHEMISTRY, VOL. 48, NO. 9, AUGUST
ters, the operator has reached the central monitor portion of the program. One of six options can be selected, each of which will normally return control to the central monitor upon completion of the option: a) Perform a background determination. b) Monitor a single mass to perform a GC peak integration. c) Monitor several masses and compute the relative abundance of each mass (isotope ratios). d) Recheck the centroid locations of all masses in the scanning range. e ) Allow the operator to reselect the masses to be monitored. f) Leave the program altogether. After initiation of option c, the program either waits for the ion flux of the most intense mass to rise above a selected threshold intensity or to rise above the background (baseline) by a selected number of standard deviations before active data acquisition begins. Using a unidirectional or bidirectional scan algorithm, each mass is observed in sequence until the ion flux decays below the stopping threshold. A real-time plot of the ion flux of the most intense mass vs. time can be displayed. After background and deadtime count loss corrections are performed, a table of masses, relative abundances, and theoretical standard deviations (based on count totals) is printed. Computer Evaluation of t h e Mass Cycling Error. Using the Indiana University CDC 6600 computer, a FORTRAN IV program was written to evaluate Equation 10, taking into account all the details mentioned in &e preceding theoretical section. The average mass cycling error, AR,can be computed as a function of a) the number of measurement cycles, b) GC peak shape, c) bidirectional or unidirectional scan algorithm, d) time delays and unequal integration times, and e) degree of chromatographic isotopic fractionation. For a given number of measurement cycles,j , the program slices up the GC peak profile into j increments, each of which is further subdivided into the times spent on mass 1and 2 plus the times spent in delays. The area per unit time of each small time slice for mass 1is summed across the GC profile anidivided by the summed area per unit time for mass 2. To obtain AR values, this procedure is repeated 20 times, incrementing the measurema cycle fraction each time (such that 9 ranges from -0.95 to 0), with AR being taken as the average of these results. The GC peak profile selected can be either a polynomial least squares fitted GC profile or a simple Gaussian curve. Both the routine for evaluating the area under a Gaussian curve and the program for performing a polynomial least squares curve fit were obtained from P. R. Bevington (7).
1976
F o r a Gaussian GC profile, the mass cycling error rapidly decreases as t h e n u m b c o f measurement cycles p e r peak increases. Figure 6 plots AR as a function of j for a Gaussian GC profile scanned either with a unidirectional or bidirectional algorithm. It has been assumed that 95% of the total peak area ( ~ Z C T is ) utilized in t h e measurement, that t i m e delays and column isotopic fractionation a r e negligible, and that the relative integration times a r e equal. T h e mass cycling error curves for a tailing GC profile ( t h e broken lines in Figure 6) will be considered later. For t h e Gaussian profile, converges from opposite directions for the unidirectional and bidirectional scans; however, the absolute magnitudes of t h e curves d o not differ dramatically. In either case, the mass cycling error is systematic-&? provides a bias away from zero (reduced accuracy) rather than a scattering effect about zero (reduced precision). (Note, however, that varying values of C$ will, in practical cases, have t h e effect of scattering observed AR values around t h e hR vs. j line.) T h e bidirectional scan generates t w o g curves-even numbers of cycles provide more negative AR than odd numbers of cycles. T h i s multiplicity can be seen clearly when an example for C$ = 0 is considered. For a bidirectional scan when C$ = 0 a n d j is odd, the GC peak is divided symmetrically a n d AR = 0. However, when C$ = 0 and j is even, t h e bidirectional scan still allows one mass to be sampled twice in succession at t h e t o p of t h e GC peak (AR < 0). In fact, with only one exception at very small j vviJued, AR < 0 when j is even for all values of 4. T h u k w h e n hR is computed, even j values produce more negative AR values than odd j values. Of course, as j increases, t h e relative differences between consecutive j values decrease, a n d t h e even and odd =curves converge. This even vs. o d d distinction cannot occur for unidirectional scanning.
a
The number of cycles G-value) required to avoid any specified average mass cycling error can be determined from Figure 6. For example, in the Gaussian case, considering a unidirectional scan, the average mass cycling error will be less than 1% whenever six or more cycles are obtained across the GC peak. Because this only specifies the average error, worse deviations can be encountered, and the number of cydes required to bring the worst case error below 1%is 9. If AR is to be held below 0.1%, 21 or more cycles will be required. T o hold the worst case error below 0.1%, 29 or more cycles will be required. A systematic consideration of observational requirements under a variety of conditions has been carried out by varying the parameters described in the preceding theoretical section and examining their individual effects. The results are summarized in Table I and discussed in the following paragraphs. Since it is almost impossible to separate a GC peak completely from the baseline, some fraction of the peak must be lost. The amount of noise in the baseline, and the initial and final slopes of the GC peak, will determine the precision with which the peak can be defined and, accordingly,what fraction of it will be lost. The effect of integrating over less than 100% of the total GC peak area is summarized in the first four entries in Table I. Especially for a unidirectional scan, a substantial reduction in the number of measurement cycles required occurs by increasing the total area of integration from 95% to 99%. Practically, this may prove difficult, as normal statistical variations in the background can falsely trigger the lowered thresholds required. However, when the total area of integration is reduced from 95% to 70%, the required number of measurement cycles is not significantly increased. No substantial penalty is paid in terms of the mass cycling error for failure to observe the entire GC peak. Changing the relative integration time, k , results in opposite effects for the mass cycling errors in unidirectional or bidirectional scanning (entries 5-8 in Table I). The average mass cycling error tends to increase with increasing k for a bidirectional scan, but to increase with decreasing k for a unidirectional scan. However, unless the value of k is extreme (around 10 or greater, corresponding to R 0.01 for optimally distributed observation times) the mass cycling error is not significantly affected by unequal integration times. The ion current ratio measurement cannot be made with perfect efficiency-some fraction of the time must be spent in switching ion beams at the detector. In many cases, the “overhead” for accelerating potential or magnetic field slewing is a non-negligible factor. If, for example, the time required to switch from mass 1 to mass 2 were the same as the time required to switch from mass 2 to mass 1,the time distributions would be:
-
Unidirectional: t l t d t 2 t d t l t d t 2 . . . Bidirectional: t l t d t 2 t 2 t d t l t d t 2 . . . where t d is the switching delay. In terms of the previously defined t d l , td2, t d 3 terminology, this example specifies t d = t d 2 for a bidirectional scan and t d = t d 2 = t d 3 for a unidirectional scan. The effect of non-negligible switching delays on the mass cycling error is illustrated in entries 11 and 13 in Table I (kdl = 0, k d s = k d 3 = 1for a unidirectional scan; kdl = k d S = 0, k d z = 1for a bidirectional scan). The number of required measurement cycles can be substantially reduced for a unidirectional scan when switching delays are present. For a bidirectional scan, a different effect is observed, with a slightly higher number of measurement cycles being required. In either case, the delays are practically “invisible” in terms of their effect on the mass cycling error. The presence of delays does not have a strong effect on the number of cycles required for any given level of accuracy.
2.0
I.o
-0
8
I$ -10
’
-20
’I
I
I I
0
’
, (0
’
2b
‘
30
s
40
.
io
Cycles/peok
Flgure 6. Calculated average mass cycling error as a function of the number of measurement cycles in any given gas chromatographic peak The solid lines represent calculations based on Gaussian peak profiles. The broken lines represent calculations based on a typical tailing GC peak. If a unidirectional scan is used, systematic errors are expected to follow one of the curves marked u. When a bidirectional scan is used, two curves are generated: one for odd (b),and one for even (6’) values of j
The consideration of delays seems to introduce a paradox: the results indicate that more accurate ratios can be obtained when less time is spent observing the ion beams. The explanation is that ion statistical considerations have been completely excluded in this work, and that the present result merely indicates that more equally distributed areas can result when delays are present. If, for example, three ratio measurements supported by arbitrarily large and equal integrated ion currents were compared, and if one of these measurements had been obtained without delays, it is expected that the acquisition of 66 measurement cycles would be required for A T < 0.01% (unidirectional scan, entry 1, Table I). On the other hand, if delays according to the pattern described above had been incorporated, the acquisition of only < 0.01% (entry 11, Table I). 45 cycles should still furnish Longer (by a factor of 5 ) delays would have a still more dramatic effect, reducing the required number of cycles to 22 (entry 12, Table I). For the sake of concreteness, it can be observed that the no-delay case calls for the peak to be divided into 132 equal time increments (66 observations on each peak, entry 1, Table I), and that, if the peak were 26 s wide, each increment would be 197 ms in length. Again considering a peak 26 s in width, the second example would call for forty-five 144-ms observations on each of two ion beams, together with ninety 144-ms delays. The third example would be represented by twenty-two 98-ms observations on each beam, together with forty-four 492-ms delays. Because we have postulated equal integrated ion currents (and, therefore, equal precision) in all three ratio measurements, but have observed the ion beams with perfect efficiency in the first example, with 50% efficiency in the second example, and with 16.6% efficiency in the third example, the ratio measurements incorporating delays would require sample amounts twice as great and six times as great, respectively, as the no-delay measurement. The most dramatic effect involving time delays occurs when more than two ion beams are measured. For example, when three masses are monitored, three ratios can be calculated:
a
ANALYTICAL’CHEMISTRY, VOL. 48, NO. 9, AUGUST 1976
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I
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mass l/mass 2, mass l/mass 3, and mass 2/mass 3. As far as the ratio mass l/mass 3 is concerned, the observation of mass 2 is totally extraneous and a waste of time. Unlike ion beam switching delays, time delays arising from the observation of extra ion beams tend to cause larger and more unequal time delays. The unidirectional scanning of five different ion beams, each observed for equal lengths of time, will, when the ratio mass l/mass 4 is considered, give rise to the measurement pattern: t l , tda, t 4 , tdb, t l , tda, t 4 , t d b . . . where tl and t 4 correspond to the times spent observing masses 1and 4 and t d a l t d b correspond to the artificial time delays created by observing the other masses: 2,3, and 5 (note that tda = t2 t3 and t d b = t 5 ) .Table I lists several different cases in which three different groupings arise. For the observation of 5 ion beams, the ratio mass 2/mass 4 involves equally spaced delays (entries 9 and 10);the ratio mass l/mass 5 includes a large time delay in the middle of the measurement cycle (entry 15); and the ratio mass Z/mass 3 involves time delays a t the beginning and end of each measurement cycle (entry 16). Of the three possibilit,ies just stated, only the last has a significant effect on the mass cycling error: the masscycling error is reduced. This effect is not unexpected. Since AR arises from measuring two ion beams a t times when the overall ion flux is changing, the use of a long delay, quick measurement of the two ion beams, and another long delay causes the two beams to be measured before the ion flux can undergo significant changes. Would it somehow be beneficial to introduce deliberate delays in order to obtain a better distribution of peak areas? In general, no. If a system is capable of switching beams fast enough to meet the no-delays cycle requirement, then the introduction of delays can serve only to degrade precision, without substantially reducing the systematic error due to mass cycling. If a given instrument control system is not capable of switching beams rapidly enough, or if many different beams must be simultaneously observed, the present calculations do show that the systematic error due to mass cycling will probably not be increased by this problem, although it is inevitable that the loss of ion-beam-observation-time will lead to a degradation of precision. Chromatographic isotopic fractionation, represented in this model by the variable 0, generates a two-stage effect on the mass cycling error. For a unidirectional scan, in the first stage, 0 # 0 allows AR to converge rapidly toward zero at small measurement cycle values but, in the second stage, causes a much slower convergence as j increases. For the bidirectional scan, AR experiences no rapid convergence to zero at small j but does undergo a very slow convergence with increasing j . Increasing 0increases the effects (see Table I, entries 17-20). If a A of -0.75% or greater can be tolerated, 0I 20% will not significantly affect the number of measurement cycles needed. Only when AR needs to be less than 0.5% will a significant change occur from the presence of fractionation. Although useful insight can be gained by assuming a Gaussian chromatographic peak profile, the effect of a skewed GC profile on AR should be considered. After xenon was injected onto the GC column from the EDF, a digital record of the ion flux for 132Xewith time was recorded, and a 9th degree polynomial least squares fit was performed in order to generate a function representing an actual GC peak. T o observe visually the amount of skew present in the GC peak, a plot of the fitted profile along with a plot of the Gaussian curve was made (Figure 7). To calculate a numerical value for the amount of skew present in the GC peak, the method of Grushka et al. (8),was applied. The skew, a measure of peak asymmetry where a full Gaussian curve would have no skew and half of a Gaussian profile (divided down the middle) would have a skew of 0.995, was calulated for the experimental GC profile to be 0.71. The excess, a measure of flattening of a peak where a Gaussian curve would have zero excess and a
+
-eo
0
+eo
Figure 7. A plot of the function fitted to an actual gas chromatographic peak. A Gaussian peak profile is shown for comparison Note that when approximately equal starting and stopping thresholds (corresponding to &2 u on the Gaussian peak) are chosen for the skewed and Gaussian profiles,a substantially smaller portion of the skewed peak area (91 % vs. 95 Yo) is enclosed by the integratlon limits
square wave would have an excess of 1.20, was calculated to be 0.31. The model calculations carried out for the Gaussian profile have been repeated for the skewed GC peak, and are summarized in Table 11. The relationship between hR a n d j in this case is given by the broken line in Figure 6. Skewing of the peak profile has the effect of reducing the distinction between vs. j curves (bidirectional scan), rethe even and odd moving the particularly favorable aspect of the odd-j case. However, GC tailing causes the unidirectional scan algorithm to produce a negative a t very 1oTmeasurement cycle values 0’ I 4)yet produce a positive U?for larger values of j . For the unidirectional scan at j > 4,a skewed GC peak gives than a symmetric Gaussian profile. In general, a smaller comparison of Tables I and I1 indicates that the change in chromatographic peak shape, in this instance, a t least, does not bring about any large changes in the data acquisition requirements. An experimental investigation of AR is of substantial importance in supporting the reality of the preceding calculations. In this regard, it must be noted that AR is defined as the relative difference between true and observed ratio values, and that, therefore, the “true” value must somehow be known. In the present context, the “true” ratio value refers to that which would be observed by a perfect detector system. The possibility of mass discrimination effects in the inlet system, ion source, and ion optical system cannot be excluded and, therefore, the “true” ratio cannot be taken from a standard compilation of isotopic abundances. Instead, the “true” ratio must be measured on the test mass spectrometer, using an ion current measurement system which somehow approximates perfection, a t least in comparison to the system being investigated. In addition, it would be useful if the true ratio could be frequently monitored, so that drifts in experimental conditions which caused the true ratio to change were not incorrectly interpreted in terms of extreme AR values. T o provide the best practicable experimental measurement of AR, a system was adopted in which diluted aliquots of xenon from the EDF were injected onto the GC column using an %port valve with the 1.0 cm3 and 0.2 cm3 sample loops.
a
a
ANALYTICAL CHEMISTRY, VOL. 48, NO. 9, AUGUST 1976
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relative standard deviations in the observed and true ratios are 0.090%and 0.022%,respectively. Accordingly, the standard deviation of any AR point is 0.09%.
n 0.0
CONCLUSIONS
- 2.0
&
1% - 4.0 ~
’
Error b a r s indicate 95 % confidence limit6 ( s a m e f o r a11 D o i n t s )
-6.0
0
20
40
60
80
I
Cycies/peak
Figure 8. Comparison between experimental data (points) and model
calculations (lines) Only calculations based on the tailing peak profile are shown. The solid lines represent the expected unidirectional (u)and bidirectional ( b )AR vs. j curves. The broken lines represent the worst- and best-case values of AR as a function of q5 at each value of j . The triangles represent unidirectional scan data: the circles, bidirectional. The indicated confidence intervals are based on ion statistlcal limitations
Repeated injections provided alternating 0.2 cm3 and 1.0 cm3 injections (with one injection about every 2 min). The ion fluxes for 131Xeand 132Xewere observed for every injection and the 131/132 xenon ratio (-0.79) was computed. For the 1.0-cm3 injections, the observation times were adjusted in order to furnish at least 260 measurement cycles, a value large enough to overcome any systematic mass cycling errors. In contrast, the observation times for the 0.2-cm3injections were adjusted to provide small numbers of cycles (4Ij I 125),so that systematic mass cycling errors could be expected. Values of AR were calculated using Equation 9, in which the “observed” ratios were those obtained from the 0.2-cm3injections and the “true” ratio was taken in each case to be the average of the six nearest ratios obtained from the l.0-cm3 injections. The experimental AR values are compared to the theoretical a?i vs. j curves (skewed peak)‘in Figure 8. It can be seen that the points lie close to the predicted lines in the range 5 < j < 15. Significantly, it is in this area that the model calculations predict the greatest errors and are, therefore, subject to the most sensitive tesLFor j > 15, the experisental points seem to scatter around AR = -0.25; not about AR = 0, as expected. This offset is probably due to pressure-dependent factors, which, by their nature, are not neutralized even in this differential measurement. The pressure in the ion source is five times greater for the l.0-cm3 injections than for the 0.2-cm3 injections. Based on the numbers of ions collected, the
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The mass cycling error, though “real” in the sense that it can be calculated and demonstrated, is, in general, rather well-behaved. That is, neither skewed peak shapes nor complicated observation patterns lead to a requirement for unusually large numbers of cycles in any given peak. For most arrangements, the mass cycling error can be kept below 1%by performing more than 10 measurement cycles per GC peak. In the absence of significant chromatographic fractionation of the isotopes ( p < 4%), the error can be kept to less than 0.1% by performing more than 30-35 cycles/peak. However, if a mass cycling error of 0.05% or lower is desired, the number of required cycles must be increased substantially and conditions of the measurement must be carefully considered. The unidirectional scanning pattern is generally superior to the bidirectional pattern in GC-MS applications, although the advantage is small enough that it might be outweighed by instrumental factors (elimination of long flyback delays, etc.) in some cases. Since modern chromatographic techniques can produce peaks only a few seconds wide, chances for significant mass cycling errors are relatively great, and, in these circumstances, very rapid beam switching can be required. On the other hand, it is unwise to cycle too rapidly when conditions do not require it. As any real system involves finite time delays for amplifier and power supply settling, rapid mass cycling only increases the fraction of each cycle spent in delays. This programmed waste of time can be minimized when the mass cycling error is also considered, and an optimum measurement scheme obtained.
ACKNOWLEDGMENT The authors thank D. R. Rushneck of Interface, Inc., for valuable discussions concerning the construction of the palladium separator GC-MS interface. We appreciate the support and encouragement of K. B. Denson, R. F. Blakely, and S. A. Studley. LITERATURE CITED A. L. Burlingame, R. E. Cox, and P. J. Derrick, Anal. Chem., 46, 248R (1974). J. M. Hayes, manuscript in preparation. D. A. Schoeller and J. M. Hayes, Anal. Chem., 47, 408 (1975). P. G. Simmonds, G. R. Shoemake, and J. E. Lovelock, Anal. Chem., 42,881 (1970). (5) J. E. Lovelock, “Gas ChromatograDhy . 1960”, R. P. W. Scott, Ed., Butterworths, London, 1960, p 26. (6) F. Bruner, P. Ciccioli, G. Crescentini, and M. Pistolesi, Anal. Cbem., 45, 1851 (1) (2) (3) (4)
(1973) \
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(7) P. R. Bevington, “Data Reduction and Error Analysis for the Physical Sciences”, McGraw-Hill, New York, N.Y., 1969. (8) E. Grushka, M. N. Meyers, P. D. Schettier, and J. C. Giddings, Anal. Chem., 41, 889 (1969).
RECEIVEDfor review March 8,1976. Accepted April 19,1976. This work was supported by the National Aeronautics and Space Administration (Grant No. NGR 15-003-118).
